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<meta name="description" content="第二章 线性表线性表：表内数据类型相同，有限序列 本章将以总结的形式展现： 2.1 顺序表与链式表的区别     顺序表 链式表     存取 随机存取 顺序存取   结构 顺序存储（连续） 随机存储（不连续）   空间分配 静态存储（可以动态分配） 动态存储   操作 查找 O(1) ,插入和删除O（n） 查找 O(n) ,插入和删除O（1）   缺点 插入删除不便，长度不可以改变 查找速度慢，">
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<meta property="og:description" content="第二章 线性表线性表：表内数据类型相同，有限序列 本章将以总结的形式展现： 2.1 顺序表与链式表的区别     顺序表 链式表     存取 随机存取 顺序存取   结构 顺序存储（连续） 随机存储（不连续）   空间分配 静态存储（可以动态分配） 动态存储   操作 查找 O(1) ,插入和删除O（n） 查找 O(n) ,插入和删除O（1）   缺点 插入删除不便，长度不可以改变 查找速度慢，">
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        <h1 class="article-title">矩阵分解与正则化</h1>
    
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        <ol class="toc"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E9%80%9A%E8%BF%87%E6%9C%AC%E6%96%87%EF%BC%8C%E4%BD%A0%E5%8F%AF%E4%BB%A5%E5%AD%A6%E5%88%B0%EF%BC%9A"><span class="toc-text">通过本文，你可以学到：</span></a></li></ol></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%89%8D%E6%8F%90%E7%9F%A5%E8%AF%86"><span class="toc-text">前提知识</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%A8%80%E7%96%8F%E7%9F%A9%E9%98%B5%EF%BC%9A"><span class="toc-text">稀疏矩阵：</span></a></li></ol></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3%EF%BC%9A"><span class="toc-text">矩阵分解：</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%AE%97%E6%B3%95%E6%AD%A5%E9%AA%A4%C2%B6"><span class="toc-text">算法步骤</span></a></li></ol></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E6%80%BB%E7%BB%93%E6%80%9D%E6%83%B3%EF%BC%9A"><span class="toc-text">总结思想：</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%B0%8F%E7%BB%93"><span class="toc-text">小结</span></a>
    
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        <h3 id="通过本文，你可以学到："><a href="#通过本文，你可以学到：" class="headerlink" title="通过本文，你可以学到："></a>通过本文，你可以学到：</h3><p>矩阵分解的损失函数</p>
<p>稀疏矩阵与稠密矩阵</p>
<p>矩阵分解</p>
<h1 id="前提知识"><a href="#前提知识" class="headerlink" title="前提知识"></a>前提知识</h1><h3 id="稀疏矩阵："><a href="#稀疏矩阵：" class="headerlink" title="稀疏矩阵："></a>稀疏矩阵：</h3><p>在矩阵中，若数值为0的元素数目远远多于非0元素的数目，并且非0元素分布没有规律时，则称该矩阵为稀疏矩阵;与之相反，若非0元素数目占大多数时，则称该矩阵为稠密矩阵。定义非零元素的总数比上矩阵所有元素的总数为矩阵的稠密度。</p>
<h1 id="矩阵分解："><a href="#矩阵分解：" class="headerlink" title="矩阵分解："></a>矩阵分解：</h1><p>1.通俗理解矩阵分解：<a target="_blank" rel="noopener" href="https://blog.csdn.net/xuejianxinokok/article/details/79230190">https://blog.csdn.net/xuejianxinokok/article/details/79230190</a></p>
<p>我们知道，要做推荐系统，最基本的一个数据就是，用户-物品的评分矩阵:</p>
<p><img src="https://gitee.com/moluggg/image/raw/master/img/202008/09/215454-20962.png" alt="biao.png"></p>
<p>矩阵中，描述了5个用户(U1,U2,U3,U4 ,U5)对4个物品(D1,D2,D3,D4)的评分(1-5分)，- 表示没有评分，现在目的是把没有评分的 给预测出来，然后按预测的分数高低，给用户进行推荐。</p>
<hr>
<p>推导开始:</p>
<p>如何预测缺失的评分呢？对于缺失的评分，可以转化为基于机器学习的回归问题，也就是连续值的预测，对于矩阵分解有如下式子，R是类似上图的评分矩阵，假设N<em>M维(N表示行数，M表示列数)，可以分解为P跟Q矩阵，其中P矩阵维度N</em>K，P矩阵维度K*M。</p>
<p><img src="https://gitee.com/moluggg/image/raw/master/img/202008/09/215456-826457.png" alt="1.png">式子1</p>
<p>对于P,Q矩阵的解释，直观上，P矩阵是N个用户对K个主题的关系，Q矩阵是K个主题跟M个物品的关系，至于K个主题具体是什么，在算法里面K是一个参数，需要调节的，通常10~100之间。</p>
<p><img src="https://gitee.com/moluggg/image/raw/master/img/202008/09/215457-594267.png" alt="2.png">式子2</p>
<p>对于式子2的左边项，表示的是R^ 第i行，第j列的元素值，对于如何衡量，我们分解的好坏呢，式子3，给出了衡量标准，也就是损失函数，平方项损失，最后的目标，就是每一个元素(非缺失值)的e(i,j)的总和 最小</p>
<p><img src="https://gitee.com/moluggg/image/raw/master/img/202008/09/215458-411457.png" alt="3.png">式子3</p>
<p>OK，目前现在评分矩阵有了，损失函数也有了，该优化算法登场了，下面式子4是基于梯度下降的优化算法，p,q里面的每个元素的更新方式</p>
<p><img src="https://gitee.com/moluggg/image/raw/master/img/202008/09/215459-38814.png" alt="4.png">式子4</p>
<p>然而，机器学习算法都喜欢加一个正则项，这里面对式子3稍作修改，得到如下式子5，beita 是正则参数</p>
<p><img src="https://gitee.com/moluggg/image/raw/master/img/202008/09/215500-709070.png" alt="5.png">式子5</p>
<p>相应的p,q矩阵各个元素的更新也换成了如下方式</p>
<p><img src="https://gitee.com/moluggg/image/raw/master/img/202008/09/215501-759298.png" alt="6.png">式子6</p>
<p>至此，P,Q矩阵元素求出来了之后，计算某个用户i对某个物品j的评分计算就是p(i,1)<em>q(1,j)+p(i,2)</em>q(2,j)+….+p(i,k)*q(k,j)。</p>
<h3 id="算法步骤¶"><a href="#算法步骤¶" class="headerlink" title="算法步骤¶"></a>算法步骤<a target="_blank" rel="noopener" href="http://docs.sqdxwz.com/ai-note/ai/mf/#_12">¶</a></h3><p>1）随机生成两个矩阵P,Q；</p>
<p>2）为了能够有较好的泛化能力，会在损失函数中加入正则项对参数进行约束；</p>
<p><img src="https://gitee.com/moluggg/image/raw/master/img/202008/09/215521-732148.png" alt="5.png"></p>
<p>3）使用梯度下降法获得修正的p和q；</p>
<p><img src="https://gitee.com/moluggg/image/raw/master/img/202008/09/215513-739043.png" alt="6.png"></p>
<p>4）不停迭代直至算法最终收敛，即：直到sum（e^2）&lt;=阈值或者达到指定的训练次数。</p>
<p><strong>注意:</strong>算法步骤中的更新公式和损失函数都已经进行了<a target="_blank" rel="noopener" href="https://baike.sogou.com/v7753108.htm?fromTitle=正则化">正则化</a>。</p>
<h1 id="总结思想："><a href="#总结思想：" class="headerlink" title="总结思想："></a>总结思想：</h1><p><strong>矩阵分解就是，将一个待求矩阵分解成两个矩阵（自定义就行），根据损失函数公式，利用梯度下降法，将两个矩阵对应位置的数据进行修改，达到能合成矩阵与原矩阵十分吻合为止。</strong></p>
<pre><code class="lang-python">
# !/usr/bin/env python
# encoding: utf-8
__author__ = &#39;Scarlett&#39;
#矩阵分解在打分预估系统中得到了成熟的发展和应用
# from pylab import *
import matplotlib.pyplot as plt
from math import pow
import numpy


def matrix_factorization(R,P,Q,K,steps=5000,alpha=0.0002,beta=0.02):
    #alph学习效率，beta是正则项系数,R是待求矩阵，P Q是分解成的矩阵
    Q=Q.T  # .T操作表示矩阵的转置
    result=[]，
    for step in range(steps):
        for i in range(len(R)):#一个矩阵的列
            for j in range(len(R[i])):#一个矩阵的行
                if R[i][j]&gt;0:#若为0，表示这个地方没有数据，即待预测的数据 
                    eij=R[i][j]-numpy.dot(P[i,:],Q[:,j])#后边表示输入P该行，该列中所有元素
                    &#39;&#39;&#39;&#39;计算误差&#39;&#39;&#39;&#39;&#39;
                    #np.dot()   一维矩阵相乘得到內积，多维就是离散课上见得方法，扫射。
                    &#39;&#39;&#39;https://zhidao.baidu.com/question/188716142.html矩阵的內积&#39;&#39;&#39;
                    for k in range(K):&#39;&#39;&#39;求导&#39;&#39;&#39;
                        P[i][k]=P[i][k]+alpha*(2*eij*Q[k][j]-beta*P[i][k])
                        Q[k][j]=Q[k][j]+alpha*(2*eij*P[i][k]-beta*Q[k][j])
        eR=numpy.dot(P,Q)
        e=0
   ##以下求出的损失函数而做的准备
        for i in range(len(R)):
            for j in range(len(R[i])):
                if R[i][j]&gt;0:
                    e=e+pow(R[i][j]-numpy.dot(P[i,:],Q[:,j]),2)
                    for k in range(K):
                        e=e+(beta/2)*(pow(P[i][k],2)+pow(Q[k][j],2))#给e加上正则项，循环次数取决于你自定义的K的大小
        result.append(e)
        if e&lt;0.001:
            break
    return P,Q.T,result

if __name__ == &#39;__main__&#39;:
    R=
        [5,3,0,1],
        [4,0,0,1],
        [1,1,0,5],
        [1,0,0,4],
        [0,1,5,4]
    ]

    R=numpy.array(R)

    N=len(R)
    M=len(R[0])
    K=2
##就这么随便的生成么？
    P=numpy.random.rand(N,K) #随机生成一个 N行 K列的矩阵
    Q=numpy.random.rand(M,K) #随机生成一个 M行 K列的矩阵

    nP,nQ,result=matrix_factorization(R,P,Q,K)
    print(&quot;原始的评分矩阵R为：\n&quot;,R)
    R_MF=numpy.dot(nP,nQ.T)
    print(&quot;经过MF算法填充0处评分值后的评分矩阵R_MF为：\n&quot;,R_MF)

#-------------损失函数的收敛曲线图---------------

    n=len(result)
    x=range(n)
    plt.plot(x,result,color=&#39;r&#39;,linewidth=3)
    plt.title(&quot;Convergence curve&quot;)
    plt.xlabel(&quot;generation&quot;)
    plt.ylabel(&quot;loss&quot;)
    plt.show()
</code></pre>
<pre><code class="lang-python">import numpy as np
import matplotlib.pyplot as plt
import math

x=np.array([0.05,0.10])#数值
y=np.array([[0.15,0.25],[0.2,0.3]])#权重
y1=np.array([[0.4,0.5],[0.45,0.55]])#第二层权重
r=np.array([0.01,0.99])#期望输出

b=0.35
b1=0.6
q=0.5#学习率

def neth(x,y,b):
    return np.dot(x,y)+b     #将权重与数值相乘
##np.dot()   一维矩阵相乘得到內积，多维就是离散课上见得方法，扫射。

def outh():

    outh=[]

    for i in range(len(y[0])):

        a=1/(math.exp(-neth(x,y,b)[i])+1)# sigmoid函数

        outh.append(a)##

    return np.array(outh)

def neto():

    return np.dot(outh(),y1)+b1

def outo():

    outo=[]

    for i in range(len(y1[0])):

        a=1/(math.exp(-neto()[i])+1)

        outo.append(a)

    return np.array(outo)
o=0
O=[]
result=[]
#梯度下降
while True:
    o=o+1
    dif=[[0,0],[0,0]]
    dif1=[[0,0],[0,0]]
      a=0
#对w5-w8进行求导
     for i in range(len(y1)):
        for j in range(len(y1[i])):
              dif1[i][j]=(-(r[j]-outo()[j]))*(outo()[j]*(1-outo()[j]))*(outh()[i])
              y1[i][j]=y1[i][j]-q*dif1[i][j]#进行权值的更新
#对w1-w4进行求导
      for i in range(len(y)):
        for j in range(len(outo())):
              a=a+(-(r[j]-outo()[j]))*outo()[j]*(1-outo()[j])*y1[i][j]
            #两个输出层就两个，对从输出层到隐含层的数值求导之和，前面公式有蓝色为底第一行前半部分
            for k in range(len(y[i])):
                  dif[i][k]=a*(outh()[k]*(1-outh()[k]))*x[i]#蓝色为底第二行
                  y[i][k]=y[i][k]-q*dif[i][k]#权值更新
      e=np.sum((r-outo())**2/2)    #总误差
      result.append(e)
      O.append(o)
      if e&lt;0.00001:
        break
plt.plot(O,result)
plt.show()
print(outo())
print(o)
print(e)
</code></pre>
<p>注：1，阈值又叫临界值，是指一个效应能够产生的最低值或最高值</p>
<p>这是矩阵分解的基础版，其实矩阵分解还有很多办法：</p>
<p>2.<a target="_blank" rel="noopener" href="https://blog.csdn.net/yanqingan/article/details/6125812">最大似然估计</a></p>
<p><a target="_blank" rel="noopener" href="https://blog.csdn.net/billbliss/article/details/78579308">SVD：可能学了线代才能理解</a></p>
<h1 id="小结"><a href="#小结" class="headerlink" title="小结"></a>小结</h1><p>1.矩阵分解能实现的原因？还不知道</p>
<p>2.梯度下降法梯度的更新pq值得过程</p>

      
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        $(".shang_box").fadeToggle();
        $(".hide_box").fadeToggle();
    }
    

</script>

<!--加入行号的高亮代码块样式-->

<!--自定义样式设置-->
<style>
    
    
    .nav {
        width: 542px;
    }
    .nav.fullscreen {
        margin-left: -542px;
    }
    .nav-left {
        width: 120px;
    }
    
    
    @media screen and (max-width: 1468px) {
        .nav {
            width: 492px;
        }
        .nav.fullscreen {
            margin-left: -492px;
        }
        .nav-left {
            width: 100px;
        }
    }
    
    
    @media screen and (max-width: 1024px) {
        .nav {
            width: 492px;
            margin-left: -492px
        }
        .nav.fullscreen {
            margin-left: 0;
        }
    }
    
    @media screen and (max-width: 426px) {
        .nav {
            width: 100%;
        }
        .nav-left {
            width: 100%;
        }
    }
    
    
    .nav-right .title-list nav a .post-title, .nav-right .title-list #local-search-result a .post-title {
        color: #383636;
    }
    
    
    .nav-right .title-list nav a .post-date, .nav-right .title-list #local-search-result a .post-date {
        color: #5e5e5f;
    }
    
    
    .nav-right nav a.hover, #local-search-result a.hover{
        background-color: #e2e0e0;
    }
    
    

    /*列表样式*/
    

    /* 背景图样式 */
    
    


    /*引用块样式*/
    

    /*文章列表背景图*/
    

    
</style>







</html>
